So what are the "must have" pets? I see a lot of people saying Unborn Valkyr, Tranquil Mechanical Yeti, Chrominus, Pandaren Water Spirit. Any that I'm missing?

There are none. Some Pets are really awesome in the tournament, but for almost every single pet there is another one out there, which can do the job almost equally as good. Of those you mention, I use only Chominius / Water Spirit in the tournament. And the fight I use them on would be also possible with snails, or porcupettes.
In the wowhead comments and the thread here are many different strategies with different pets for the fights in the tournament.

You don't need too many but you need the right ones for the fights, especially for 4 celestials. For example when I started doing the tournament I had around 40 max-level pets but a lot of them were useless so I still needed to level ~15 additional pets

Shademaster Kiryn isn't up this week, but food for thought to other people.

Crimson Geode (H/S breed) or Elementium Geode (to a lesser extent; it has much lower attack power) can murderize Nairn and Stormoen. You need a Geode with 273+ Speed or this doesn't work too well. Use Spark, Amplify Magic, and Elementium Bolt.

As long as Stormoen is in play keep using Spark; it'll very likely die long before you do (should take no more than three turns thanks to Spark and Call Lightning). Once Nairn is back in repeat the previous steps. If Nairn survives the second Elementium Bolt (it shouldn't, but things happen) switch the Geode to your designated pet for Summer.

If everything goes well Nairn will get killed by the Geode as well; use your usual pet for the third if your Geode is still alive (I use Darkmoon Zeppelin-- Missile, Decoy). Third pet can be whatever you want, but keep in mind it should be a backup in case the Geode dies too early.

This is really nothing more than a different way to try and keep Nairn out of the fight as much as possible. I used to do this with Kun-Lai Runt and Deep Freeze, but Crimson Geode can at least eat Stormoen for breakfast on top of it.

Originally Posted by braeldiil

The "you can do it with 5 pets" was aimed at people who were first starting the tournament. It gives you a small set to work towards, which is a lot easier than listing out a bunch of different pets to level. It's not how I do those fights anymore (well, I have multiple howlbomb teams, but they don't share pets), but for the people who are scraping by on pets knowing the absolute minimum to do the fights is really valuable.

There's no reason to do it with five pets considering you're always at the very least going to have fifteen.

A lot of creative things can happen even with suboptimal breeds if you sit down and think about it.

Hotfix on December 2:
•The Celestials (Chi-Ji, Niuzao, Xuen, and Yu'lon) have their health reduced to 25% of original health. If more than 10 players from the same faction are engaged with the Celestial, their health will scale up. The health increase will cap at 100% of their original health with 40 players.

Hotfix on December 2:
•The Celestials (Chi-Ji, Niuzao, Xuen, and Yu'lon) have their health reduced to 25% of original health. If more than 10 players from the same faction are engaged with the Celestial, their health will scale up. The health increase will cap at 100% of their original health with 40 players.